Intersection Theory
Spring 2008

Instructor: Valentina Kiritchenko e-mail: valentina dot kiritchenko at hcm dot uni-bonn dot de Office: Research 1, 125; Phone: -3184

Program:

• Intersection indices of subvarieties; degree of affine and projective varieties; divisors, self-intersection index of a divisor
• Bezout theorem for projective spaces
• Kleiman transversality theorem
• Intersection product and Chow ring of a variety; relation between Chow and cohomology rings
• Examples: intersection theory on projective spaces, Grassmannians, toric and flag varieties; Schubert calculus and enumerative geometry

• S. Kleiman, D. Laksov, Schubert Calculus, Amer. Math. Monthly, 79 (1972), pp. 1061-1082
  A nice elementary introduction to intersection theory on Grassmannians
• I. R. Shafarevich, Basic algebraic geometry. 1. Varieties in projective space. Second edition. Springer-Verlag, Berlin, 1994; Chapter on intersection indices Introduction to intersection theory from an algebraic viewpoint
• Ph. Griffiths, J. Harris, Principles of algebraic geometry. Reprint of the 1978 original. Wiley Classics Library. John Wiley and Sons, Inc., New York, 1994
  An exellent very well written book on complex algebraic geometry. A geometric rather than algebraic approach is used. Contains some intersection theory.
• S. Kleiman, Problem 15: Rigorous foundation of Schubert's enumerative calculus, Math- ematical Developments arising from Hilbert Problems, Proc. Symp. Pure Math., 28, Amer. Math. Soc. (1976) pp. 445-482
  A survey on the history of intersection theory
• L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci. SMF/AMS Texts and Monographs 6, 2001; Chapter on Grassmannians and flag varieties
  Contains Schubert calculus on Grassmannians and flag varieties.
• W. Fulton, Intersection theory, Second edition. Springer-Verlag, Berlin, 1998
  Covers all intersection theory. Very advanced.